Platonic Solids

35 34 fIve truncatIonS off with their corners Truncate the Platonic Solids to produce the five equaledged Archimedean polyhedra shown here. These truncated solids are the perfect demonstration of the Platonic Solids vertex figures triangular for the tetrahedron, cube and dodecahedron, square for the octahedron and pentagonal for the icosahedron. Each Archimedean Solid has one circumsphere and one midsphere. They have an insphere for each type of face, the larger faces having the smaller inspheres touching their centres. Each truncated solid therefore defines four concentric spheres. The five truncated solids can each sit neatly inside both their original Platonic Solid and that Solids dual. For example the truncated cube can rest its octagonal faces within a cube or its triangular faces within an octahedron. The truncated octahedron is the only Archimedean Solid that can fill space with identical copies of itself leaving no gaps. It also conceals a less obvious secret. Joining the ends of one of its edges to its centre produces a central angle which is the same as the acute angle in the famous Pythagorean 3 4 5 triangle, beloved of ancient Egyptian masons for defining a rightangle.